Estimation of electrical noise is important in the design of communication circuits. Noise has a direct effect on critical specifications like SNR (signal to noise ratio) and BER (bit error rate) that must be met in designs. Due to the complexity of most modem circuit designs, computer-aided analysis using numerical techniques is the only practical means of predicting noise performance.
Existing algorithms for noise computation in nonlinear circuits require computation and memory that scale quadratically or worse with the number of nonlinear devices. Until recently, this was not a limiting factor because traditional high-frequency communication circuitry has been composed largely of linear elements and a few nonlinear devices. The present trend towards integration of RF circuitry is, however, reversing this paradigm. With IC techniques being applied to the design of on-chip RF circuitry, large numbers of nonlinear devices are being used and purely linear elements are relatively few. Existing algorithms are impractically expensive for nonlinear noise computation in such circuits.
Recently, efficient harmonic-balance algorithms for finding the steady state of large nonlinear circuits were proposed independently by Rosch and others. (See, e.g., M. Rosch and K. J. Antreich, Schnell stationare Simulation nichtlinearer Schaltungen im Frequenzbereich, AEU, 46(3):168-176, 1992 and R. C. Melville, P. Feldmann, and J. Roychowdhury, Efficient Multi-tone Distortion Analysis of Analog Integrated Circuits, Proc. IEEE CICC, May 1995.) The algorithms achieve almost-linear performance by decomposing the harmonic balance jacobian matrix into simpler matrices that can be applied efficiently. This development alone does not, however, solve the problem of calculating nonlinear noise, because existing noise formulations cannot take advantage of the efficient harmonic balance decomposition.
Noise analysis in most circuit simulators has concentrated on stationary noise, motivated by the near-linear operation of communication amplifiers about their DC operating points. Existing algorithms for stationary noise (see, e.g., R. Rohrer, L. Nagel, R. Meyer, and L. Weber, Computationally efficient electronic-circuit noise calculation, IEEE J. Solid-State Ckts., SC-6(3):204-213, 1971) are efficient and widely used, as, for instance in the SPICE computer program. (See Thomas L. Quarles, SPICE 3C.1 User's Guide, University of California, Berkeley, EECS Industrial Liaison Program, University of California, Berkeley Calif. 94720, April 1989.)
For circuits such as mixers that are designed for nonlinear operation, stationary noise analysis does not suffice. Prediction of important phenomena such as noise up/down-conversion requires more general statistical models for noise. Two equivalent statistical formulations exist:
1. nonstationary or cyclostationary stochastic processes in time, and PA1 2. stochastic functions of frequency.
Several approaches to analyzing nonlinear noise in circuits use the second model or a discrete version. (See, e.g., S. Heinen, J. Kunisch, and L. Wolff, A Unified Framework for Computer-Aided Noise Analysis of Linear and Nonlinear Microwave Circuits, IEEE Trans. MTT, 39(12):2170-2175, December 1991). These approaches do not consider issues of stationarity explicitly, but recognize that correlations can exist between different frequency components of the frequency-domain stochastic "process". It can be shown that correlation between different frequency components of the frequency-domain stochastic model is equivalent to nonstationarity or cyclostationarity of the time-domain stochastic model.
Rizzoli et al. (V. Rizzoli, F. Mastri, and D. Masotti, General Noise Analysis of Nonlinear Microwave Circuits by the Piecewise Harmonic-Balance Technique, IEEE Trans. MTT, 42(5):807-819, May 1994) propose a general algorithm for calculating noise in nonlinear circuits using harmonic balance. Noise is treated as a stochastic function of frequency in their work, extending earlier analyses of diode mixer circuits. Rizzoli et al. formulate the noise algorithm in terms of the piecewise harmonic balance algorithm. Heinen et al. (see above) extend this analysis to the MNA circuit equation formulation.
The stochastic function of frequency noise model has the disadvantage of being mathematically cumbersome in its application to circuit equations. Since Fourier transforms do not exist for almost all samples of noise, suitable limiting arguments need to be used. This has led to a lack of rigor in some approaches adopting this model. An approach that combines rigor with relative simplicity is to compute two-dimensional/time-varying PSD (power spectral densities) or autocorrelation functions for noise within the circuit. Strom and Signell (Analysis of Periodically Switched Linear Circuits, IEEE Trans. Ckts. Syst., CAS-24(10):531-541, October 1977) apply these concepts to switched linear circuits to obtain closed-form expressions for computer implementation. Using time-varying and two-dimensional spectral density functions, they obtain expressions for the output noise of a switched circuit excited by stationary noise sources. They also proved that low-pass filtering of cyclostationary noise results in stationary output noise. Hull and Meyer (A Systematic Approach to the Analysis of Noise in Mixers, IEEE Trans. Ckts. Syst.-I: Fund. Th. Appl., 40(12):909-919, December 1993) base their noise analysis on frequency shifting of stationary PSD functions, without explicitly considering cyclostationary effects. However, they use the fact that lowpass filtering restores stationarity in their analysis. Okumura et al. (M. Okumura, H. Tanimoto, T. Itakura, and T. Sugawara, Numerical Noise Analysis for Nonlinear Circuits with a Periodic Large Signal Excitation Including Cyclostationawy Noise Sources, IEEE Trans. Ckts. Syst.-I: Fund. Th. Appl., 40(9):581-590, September 1993) use time-varying PSD concepts to formulate an essentially harmonic balance algorithm for noise computation. Their model for cyclostationary noise inputs, however, consists of a sum of pulse-function weighted independent stationary processes, which assumes short-term correlations and therefore does not extend to modulated flicker noise. Nevertheless, they obtain a valid expression for average output noise power and verify their algorithm with measurements.
The above approaches all use the periodicity of the steady state to justify assumptions about the cyclostationarity of noise. A more general approach using time-varying autocorrelation functions, but without the restriction of cyclostationarity, is taken by Dernir. (A. Demir, Time Domain Non Monte Carlo Noise Simulation For Nonlinear Dynamic Circuits with Arbitrary Excitations, Technical Report Memorandum No. UCI3/ERL M94/39, EECS Dept., Univ. Calif. Berkeley, 1994.) In this approach, a sequence of stochastic linear time-varying differential equations (SDEs) is derived for the circuit excited by white noise sources. By solving these SDEs numerically in the time domain, nonstationary and transient second-order statistics of the noise can be calculated. Although this approach is limited technically to white noise inputs, colored input noise (e.g., flicker noise) is treated using artificial PSD shaping networks within the circuit.
While the methods mentioned above are useful for a large class of microwave circuits, their computation and memory requirements grow at least quadratically as a function of nonlinear device count. They are therefore often impractical for analyzing integrated RF circuits, which typically contain many nonlinear devices.